Question

If $f(x) = x^2 + ax + b$ has minima at $x = 3$ whose value is $5$, then the values of $a$ and $b$ are respectively

• A. $-6, -14$
• B. $-6, 14$
• C. 14, $-6$
• D. 6, 14

Hint:

The vertex of any upward-opening parabola $f(x) = x^2 + ax + b$ is its minimum point, located at $x = -\frac{a}{2}$. Given that the minimum is at $x = 3$, we have $-\frac{a}{2} = 3 \implies a = -6$. This immediately eliminates choices (C) and (D)!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Given f(x) = x² + ax + b has a minimum value of 5 at x = 3, we must find the constants a and b.

Step 2: Key Formula or Approach:
At a minimum, f'(c) = 0 and f(c) gives the minimum value.

Step 3: Detailed Explanation:
f'(x) = 2x + a. f'(3) = 0 → 6 + a = 0 → a = –6. Then f(3) = 5 → 9 – 18 + b = 5 → b = 14.

Step 4: Final Answer:
The values are a = –6 and b = 14, matching option (B).